Introduction to Probability and Statistics for Engineers and Scientists

10.3One-Way Analysis of Variance 443

Since there are a total ofnmindependent normal random variablesXij, it follows that
the sum of the squares of their standardized versions will be a chi-square random variable
withnmdegrees of freedom. That is,

∑m

i= 1

∑n

j= 1

(Xij−E[Xij])^2 /σ^2 =

∑m

i= 1

∑n

j= 1

(Xij−μi)^2 /σ^2 ∼χnm^2 (10.3.1)

To obtain estimators for themunknown parametersμ 1 ,...,μm, letXi. denote the
average of all the elements in samplei; that is,

Xi=

∑n

j= 1

Xij/n

The variableXi. is the sample mean of theith population, and as such is the estimator of
the population meanμi, fori=1,...,m. Hence, if in Equation 10.3.1 we substitute
the estimatorsXi. for the meansμi, fori=1,...,m, then the resulting variable

∑m

i= 1

∑n

j= 1

(Xij−Xi.)^2 /σ^2 (10.3.2)

will have a chi-square distribution withnm−mdegrees of freedom. (Recall that 1 degree
of freedom is lost for each parameter that is estimated.) Let

SSW=

∑m

i= 1

∑n

j= 1

(Xij−Xi)^2

and so the variable in Equation 10.4 isSSW/σ^2. Because the expected value of a chi-
square random variable is equal to its number of degrees of freedom, it follows upon
taking the expectation of the variable in 10.4 that

E[SSW]/σ^2 =nm−m

or, equivalently,

E[SSW/(nm−m)]=σ^2

We thus have our first estimator ofσ^2 , namely,SSW/(nm−m). Also, note that this
estimator was obtained without assuming anything about the truth or falsity of the null
hypothesis.